On a conjecture of Sun

TL;DR

This paper proves conjectures by Z.-H. Sun relating representations of integers as quadratic forms and triangular forms using theta function identities, and identifies new triplets satisfying these relations.

Contribution

The paper confirms Sun's conjectures on the relation between quadratic and triangular representations and introduces new triplets that satisfy these conjectures.

Findings

Proved Sun's conjectures using theta function identities.

Established relations between quadratic and triangular representations.

Discovered new triplets satisfying the conjectures.

Abstract

A number of the form $x(x+1)/2$ where $x$ is an integer is called a triangular number. Suppose, $N(a_1,\cdots,a_k;n)$ and $T(a_1,\cdots,a_k;n)$ denote the number of ways $n$ can be expressed as $\sum_{i=1}^k a_ix_i^2$ and $\sum_{i=1}^k a_i\frac{x_i(x_i+1)}{2}$, respectively. Z.-H. Sun, in \cite{4}, conjectured some relations between $T(a,b,c;n)$ and $N(a,b,c;8n+a+b+c)$. In this paper, we prove these conjectures using theta function identities. Moreover, we add some new triplets $(a,b,c)$ satisfying these conjectures.